"stitched" style panorama in Blender? (NOT equirectangular)
If you want a spherical perspective as your link suggests, then fisheye equisolid is the type of camera you want to use. You can find it by activating the Cycles render engine, then selecting Panoramic in the Camera type and then Fisheye Equisolid in the panorama type.

You can also do it the "post-prod" way by first rendering using a rectilinear camera (called "Perspective" in Blender) because it renders faster and has a 1 on 1 correspondence with your viewport, and then use the "Lens distortion" node in the compositor. However, this one gives an elliptical distortion if your image is a rectangle (just like a Fisheye Equidistant camera). You'd have to set up a node tree to get the real spherical distortion. It could be done using a displacement node; some gradient textures and some math nodes, but you may get some subtle sub-pixel artifacts due to discontinuities in the gradient texture node.
My web search didn't give me any result for someone who already set up a node tree for that particular projection but maybe I missed it.